document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197652>Question 1197652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The growth model for a population of bacteria is given by P(t)=35e^2t where t is measured in days. How many days will it take until the number of bacteria has exceeded 17,500? </SPAN>\n" );
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document.write( "A useful saying: \"If the variable is in the trees, then log it down\". By \"trees\", I'm referring to exponents.\r<BR>\n" );
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document.write( "We'll use logarithms to isolate the exponent in the variable, so we can isolate the variable itself.\r<BR>\n" );
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document.write( "With that in mind, we replace P(t) with 17500 and solve like so.<BR>\n" );
document.write( "P(t) = 35e^(2t)<BR>\n" );
document.write( "17500 = 35e^(2t)<BR>\n" );
document.write( "17500/35 = e^(2t)<BR>\n" );
document.write( "500 = e^(2t)<BR>\n" );
document.write( "Ln(500) = Ln( e^(2t) ) ... this is where logs come in<BR>\n" );
document.write( "Ln(500) = 2t*Ln(e) .... use the rule log(A^B) = B*log(A)<BR>\n" );
document.write( "Ln(500) = 2t*1<BR>\n" );
document.write( "2t = Ln(500)<BR>\n" );
document.write( "t = Ln(500)/2<BR>\n" );
document.write( "t = 3.1073040492111<BR>\n" );
document.write( "It will take a little over 3 days to reach a population of 17,500 bacteria.\r<BR>\n" );
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document.write( "For day 0, aka the starting day, we have:<BR>\n" );
document.write( "P(t) = 35e^(2t)<BR>\n" );
document.write( "P(0) = 35e^(2*0)<BR>\n" );
document.write( "P(0) = 35e^(0)<BR>\n" );
document.write( "P(0) = 35*1<BR>\n" );
document.write( "P(0) = 35<BR>\n" );
document.write( "There are 35 bacteria at the start.\r<BR>\n" );
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document.write( "Let's see how many there are for day 1.<BR>\n" );
document.write( "P(t) = 35e^(2t)<BR>\n" );
document.write( "P(1) = 35e^(2*1)<BR>\n" );
document.write( "P(1) = 35e^(2)<BR>\n" );
document.write( "P(1) = 35*7.38905609893584<BR>\n" );
document.write( "P(1) = 258.616963462754<BR>\n" );
document.write( "So about 258 bacteria if we only consider whole numbers of bacteria.\r<BR>\n" );
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document.write( "Now day 3<BR>\n" );
document.write( "P(t) = 35e^(2t)<BR>\n" );
document.write( "P(3) = 35e^(2*3)<BR>\n" );
document.write( "P(3) = 35e^(6)<BR>\n" );
document.write( "P(3) = 35*403.428793493586<BR>\n" );
document.write( "P(3) = 14,120.0077722756<BR>\n" );
document.write( "About 14120 bacteria at this point. We're getting pretty close to the target 17,500.\r<BR>\n" );
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document.write( "Let's look at day 4<BR>\n" );
document.write( "P(t) = 35e^(2t)<BR>\n" );
document.write( "P(4) = 35e^(2*4)<BR>\n" );
document.write( "P(4) = 35e^(8)<BR>\n" );
document.write( "P(4) = 35*2980.9579870501<BR>\n" );
document.write( "P(4) = 104,333.529546753<BR>\n" );
document.write( "We've reached about 104333 bacteria which is definitely way over the target of 17,500.\r<BR>\n" );
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document.write( "These numeric results help show that we reach a population of 17,500 in the interval 3 < t < 4, which helps confirm the answer we reached a while back earlier.\r<BR>\n" );
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document.write( "Answer: Approximately 3.107304 days.<BR>\n" );
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